Question

the radius and slant height of a cone are in the ratio 8:17 if it's curved surface area is 54πcm2 then find the volume ​

Answer 1

Hello there!

The radius ($r$) and slant height ($l$)are in the ratio of 8:17.

Let's take $r=8x$, and $l=17x$.

The CSA is given as $54\pi cm^{2}$.

$CSA=\pi rl$

$54\pi=\pi rl$

We can cancel the $\pi$:

$54=rl$

$54=8x \times 17x$

$54=136x^{2}$

$x^{2}=\frac{54}{136}$

$x=\sqrt{\frac{27}{68}}$

Radius:

$r=8x$

$r=8 \times \sqrt{\frac{27}{68}}$

$r \approx 5.041$

Slant height:

$l=17x$

$l=17 \times \sqrt{\frac{27}{68}}$

$l \approx 10.712$

We must also find the height to find the volume.

$h^{2}=l^{2}-r^{2}$

$h=\sqrt{l^{2}-r^{2}}$

$h=\sqrt{10.712^{2}-5.041^{2}}$

$h \approx 9.451$

Now, we need to find the volume, which is $\frac{1}{3}\pi r^{2} h$

$V=\frac{1}{3}\pi r^{2} h$

$r \approx 5.041$ and $h \approx 9.451$

$V=\frac{1}{3}\pi \times 5.041^{2} \times 9.451$

$V=\pi \times 25.411 \times 3.15$

$V=251.467 cm^{3}$

Therefore, the volume is 251.467 $cm^{3}$.

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